L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.707 − 0.707i)8-s + (0.965 − 0.258i)9-s + (1.12 − 1.46i)11-s + (0.500 − 0.866i)16-s + (0.499 + 0.866i)18-s + (1.70 + 0.707i)22-s + (1.36 − 0.366i)23-s + (−0.965 − 0.258i)25-s + (−0.707 + 1.70i)29-s + (0.965 + 0.258i)32-s + (−0.707 + 0.707i)36-s + (−0.0999 + 0.758i)37-s + (−0.292 − 0.707i)43-s + (−0.241 + 1.83i)44-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.707 − 0.707i)8-s + (0.965 − 0.258i)9-s + (1.12 − 1.46i)11-s + (0.500 − 0.866i)16-s + (0.499 + 0.866i)18-s + (1.70 + 0.707i)22-s + (1.36 − 0.366i)23-s + (−0.965 − 0.258i)25-s + (−0.707 + 1.70i)29-s + (0.965 + 0.258i)32-s + (−0.707 + 0.707i)36-s + (−0.0999 + 0.758i)37-s + (−0.292 − 0.707i)43-s + (−0.241 + 1.83i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.280957588\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.280957588\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.965 + 0.258i)T^{2} \) |
| 5 | \( 1 + (0.965 + 0.258i)T^{2} \) |
| 11 | \( 1 + (-1.12 + 1.46i)T + (-0.258 - 0.965i)T^{2} \) |
| 13 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.258 + 0.965i)T^{2} \) |
| 23 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.0999 - 0.758i)T + (-0.965 - 0.258i)T^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.465 - 0.607i)T + (-0.258 - 0.965i)T^{2} \) |
| 59 | \( 1 + (-0.258 - 0.965i)T^{2} \) |
| 61 | \( 1 + (0.258 - 0.965i)T^{2} \) |
| 67 | \( 1 + (0.758 - 0.0999i)T + (0.965 - 0.258i)T^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.341255210860866679576329769159, −8.937677909035605305901842537539, −8.113961794057146236712684611177, −7.11510345920967034734913939759, −6.60326945495925836457551768876, −5.78304605084551903314748723365, −4.86205494055028370454578404327, −3.87850318134142175550693680608, −3.22950628708172012774376965815, −1.20258403451416158438986234569,
1.43890372712622404381595271969, 2.22596918028077293939360721292, 3.65832458456491740958238174288, 4.32091839715597638296715882756, 5.04508632563340569865273566480, 6.19025515468313009349656577204, 7.14457367000993295293657947190, 7.903583966486814698170445921189, 9.235612918639542493764515952061, 9.528095673270649420172139959514